Use the method of this example to calculate F · dr, C where F (x, y) = 2xyi + (y2 - x2) j (x2 + y2) 2 and C is any positively oriented simple closed curve that encloses the origin.

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Mathematics

Elle

14 May, 16:52

Use the method of this example to calculate F · dr, C where F (x, y) = 2xyi + (y2 - x2) j (x2 + y2) 2 and C is any positively oriented simple closed curve that encloses the origin.

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Bridger Barrett · Answer 1

Using Green's theorem we have:

int F. dr = int int d/dx ((y^2-x^2) / (x^2+y^2) ^2) - d/dy (2xy / (x^2+y^2) ^2) =

(2x^3 - 6xy^2) / (x^2+y^2) ^3 - (2x^3 - 6xy^2) / (x^2+y^2) ^3 = 0

Therefore:

int F. dr = 0